NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 47 - 54 of 101

To evaluate

Take

Here is a pole of order 1 and lies within the square.

These are simple poles which lie outside the square

So they will not contribute towards the residue of

By Cauchy residue theorem

Now

So this is form (Exponential form)

Now using the result

Or if

Then

… eq. (1)

form

From (1)

is the contour shown in the figure

To evaluate

Now is a pole of order 1 (simple pole)

Also lies within C

=

By Cauchy residue theorem

Here

Put

This is the required Taylor expansion of f in the neighbourhood of

Option (a) The given series is

Where

Now comparing it with

1)

2)

3)

By Leibnitz test (alternating series test)

The given series is convergent

Which is finite & non – zero

converge or diverge together

But is divergent by p – test

is divergent (By comparison test)

is not absolutely convergent

is conditionally convergent

Option (b) The given series i…

Let the omen series be denoted by … eq. (1)

Where

Since for absolute convergence, we take

Now is convergent if … eq. (2)

And divergent if (By p-test)

Also eq. (1) is convergent for by Leibnitz test

is convergent by Leibnitz test for … eq. (3)

From eq. (2) & (3)

We get eq. (1) is convergent